Complex physical systems, such as musical instruments, are difficult to physically reproduce or replicate faithfully in terms of their structures, minute variations of which can result in observably different response characteristics. For such systems, it may be preferable to mathematically model the behavior of the system and sythesize the output based on the model; that is, the model becomes (or operates) as a replacement for the physical system itself.
Simple physical systems (or more complex systems treated simply for purposes of approximation) of course can be straightforwardly "modeled" through direct use of the physical parameters that govern the system's behavior; for example, the motion of a simple pendulum can be precisely characterized (ignoring air resistance) in terms of length and angular acceleration, or the vibration of a string in terms of amplitude and angular frequency. Far less tractable is the goal of constructing, for example, an electronic violin capable of emulating a Stradivarius with any degree of fidelity. Complex systems contain tremendous numbers of degrees of freedom; accordingly, "first principles" physical modeling of their responses is highly difficult, both in terms of computational requirements--to keep up with changes in the many degrees of freedom--and sufficiently precise measurements of their values. Not surprisingly, approaches to physical modeling require significant approximation. For example, European Patent Application No. 0583043 ("Tone Generation System") describes the use of digital waveguides to model the behavior of musical instruments. The modeled waveforms do not represent underlying degrees of freedom in any rigorous sense, and are not related in any physically meaningful way to system inputs.
A more computationally tractable approach involves approximation or sampling techniques, whereby, for example, a list of known inputs and observed outputs is constructed, and behavior "modeled" through lookup and interpolation among entries; see, e.g., U.S. Pat. No. 5,521,322. This approach is limited, however, by the coarseness of the entries and the constrained generality of any model unconnected with the underlying physics of the system. Thus, musical instruments have been synthesized by storing output sounds for numerous known inputs--i.e., specific player manipulations of the actual instrument--and the instrument synthesized through interpolation among the sampled sound based on provided input. In practice, while short segments of such recorded sounds can be faithful, the overall response to the player's actions is not.
A related approach, exemplified by the disclosure of U.S. Pat. No. 4,018,121 (to Chowning) is to model the output behavior of a physical system as a mathematical waveform, without reference to the underlying physical degrees of freedom. Although output modeling may reflect greater attention to actual system behavior than a mechanistic interpolation approach, the validity of the output model, once again, is ultimately limited due to the absence of any connection with the true system degrees of freedom.
The "first principles" approach can be made more tractable by recognizing that not all possible physical degrees of freedom participate in the generation of a system output; a faithful model of a Stradivarius, for example, does not require detailed knowledge at the molecular level. Thus, machine learning and state-space reconstruction have been employed as intermediate approaches between a complete but computationally unachievable specification of a physical system, on one hand, and purely numerical techniques without reference to system state on the other. State-space reconstruction by the technique of time-delay embedding permits recovery and modeling of those effective physical degrees of freedom relevant to observed behavior; that is, the part of the "configuration space" (which specifies the values of all potentially accessible physical degrees of freedom) that the system actually explores as its dynamics unfolds. In accordance with the technique of time-delay embedding, a time series is measured from such a system, and the entries used to define a lag vector. For example, if s is the state vector describing the system (in terms of effective degrees of freedom rather than the complete configuration space), ds/dt=.function.(s) denotes the effective governing equations, and the measured quantity observed over time is y=y(s(t)) (where y is a vector or a scalar quantity such as amplitude or temperature), then given a delay time .tau. and a dimension d, a lag vector x may be defined as x=(y.sub.t,y.sub.t-.tau., . . . ,y.sub.t-(d-1).tau.). The central result of time-delay embedding is that the behavior of .function.(s) and x will differ only by a smooth, local, invertible change of coordinates for almost every possible choice of s, y(s) and .tau., so long as d (the "embedding dimension," i.e., the number of time lags) is sufficiently large, y depends on at least some of the components of s, and the remaining components of s are coupled by the governing equations to the ones that influence y; this result can be generalized to linear transformations on lag vectors as well.
In other words, if an experimentally observed quantity arises from deterministic governing equations, it is possible to use time-delay embedding to recover a representation of the relevant internal degrees of freedom of the system from the observable; and because the relevant degrees of freedom typically are a relatively small subset of the configuration space, the solution, while highly accurate, is also computationally tractable. System behavior can thus be modeled based on the mapping between the lag vector, whose time-varying components are measurable, and internal (and therefore generally inaccessible) system states.
In a driven system, some user input u is imposed on the system and affects its dynamic behavior; for example, the system might be a violin and the input the player's drawing of the bow. In this case, while it would be desirable to predict the behavior of the system from the input rather than the observable or the inaccessible internal degrees of freedom, this is generally not possible. Put differently, the evolution of the system cannot be described simply as y=.function.(u), since the system's behavior depends on its prior history as well as the input; the system is said to have "memory." If x is the embedding vector--which, again, maps smoothly to internal system behavior and therefore provides a complete specification of the internal degrees of freedom relevant to the observable--then the behavior of the system can be modeled given knowledge of the embedding vector x, which includes time lags on both y and u. Accordingly, it is possible to use time-delay embedding to predict y from time lags on y and u, i.e., the history of the output and the input. For example, in the above-noted case of a violin, y is amplitude (a scalar quantity) and u is the time-varying bow position and/or fingering. With x characterized, a function y=.function.(x,u) can be derived to model and predict system behavior in terms of the unobservable degrees of freedom (modeled by x) and the observable input u.
The variables y and x are related by an unknown joint probability density p(y,x) (which for a deterministic system reduces to a prediction surface y(x), while for a stochastic system samples can be drawn from p(y,x) to emulate the system's behavior). Given a set of experimental measurements {y.sub.n,x.sub.n }.sub.n=1.sup.N, the goal of data analysis is to infer a model that can predict the output or its distribution from a measurement of a new input, and to characterize the relationship between y and x. Since there is rarely enough data to estimate the unconditional density p(y,x), it is more common to seek conditional quantities such as the likelihood y.vertline.xand the error .sigma..sub.y.sup.2 .vertline.x. Traditionally these have been written in the form ##EQU1## where the .beta..sub.m quantities are unknown linear coefficients and the f.sub.m quantities are known basis functions (such as polynomial expansions). More powerful techniques, such as neural networks, utilize coefficients inside nonlinear basis functions of the form ##EQU2##
Any phenomenological model must balance underfitting (in which the model is not flexible enough to describe the data) and overfitting (in which the model is so flexible that it describes noise in the data that does not generalize). While this has been done by varying the number M of basis functions f.sub.m, such an approach is rarely justified because globally simple behavior might require a large number of terms to be represented in a given basis.